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PERSISTENCE AND transversality OF HOMOCLINIC ORBITS FOR DIFFEOMORPHISMS

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Annals of Differential Equations 2002年第4期18卷 330-343页

作者：沙罗金朱德明

A generalized method combining the exponential dichotomy and the theory of transversality was used to give conditions for the persistence and transversality of homoclinic orbits under small perturbation for the diffeo... 详细信息

关键词： exponential dichotomy invariant manifold homoclinic manifold Mel-nikov vector transversality

Melnikov Vector and Heteroclinic Manifolds

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Science China Mathematics 1994年第6期37卷 673-682页

作者：朱德明 Department of Mathematics East China Normal UniversityShanghai 200062PRC

Using the exponential dichotomies,the transversality theory and the generalized Melnikov method,we consider the conditions for the persistence and the transversality of the singular orbit,with high degeneracy,situated on the heteroclinic or homoclinic manifold under *** results obtained extend,include and improve the corresponding ones given in certain papers well known in this area.

关键词： invariant manifold heteroclinic orbit homoclinic orbit transversality Melnikov vector

Another proof for C^1 stability conjecture for flows

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Science China Mathematics 1998年第10期41卷 1076-1082页

作者：甘少波 Department of Mathematics Peking University Beijing China

The C 1 structural stability conjecture for flows by C 1 connecting lemma and obstruction sets is proved.

关键词： axiom A strong transversality obstruction set stability conjecture.

On Initial Data of the Monopole Equation

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Acta Mathematica Sinica,English Series 2009年第12期25卷 2127-2132页

作者： Hong Yu WANG Xiu Juan ZHU School of Mathematical Sciences Yangzhou University Yangzhou 225002 P. R. China

The space-time monopole equation is the reduction of anti-self-dual Yang-Mills equations in R2,2 to R2,1. This equation is a non-linear wave equation, and can be encoded in a Lax pair. An equivalent Lax pair is used by Dai and Terng to construct monopoles with continuous scattering data, and then the equation can be linearized by the scattering data, allowing one to use the inverse scattering method to solve the Cauchy problem with rapidly decaying small initial data. In this paper, we use the terminology of holomorphic bundle and transversality of certain maps, parametrized by initial data, to give more initial data, with which we can use scattering method to solve the Cauchy problem of the monopole equation up to gauge transformation.

关键词： continuous scattering holomorphically trivial bundle transversality

(r, s)-STABILITY OF UNFOLDING OF г-EQUIVARIANT BIFURCATION PROBLEM

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Acta Mathematica Scientia 2005年第3期25卷 408-418页

作者：刘恒兴张敦穆 School of Mathematics and Statistics Wuhan University

In this paper,the (?)-equivariant (s, t)-equivalence relation and (?)-equivariant infinitesimally (r, s)-stability of (?)-equivariant bifurcation problem are defined. The criterion for (?)-equivariant infinitesimally (r, s)-stability is proven when (?) is a compact finite Lie group .transversality condition is used to characterize the stability.

关键词： г-equivariant bifurcation problem unfolding transversality г-equivariant infinitesimally (r s)-stability г-equivariant (r s)-equivalence.

Orbits Heteroclinic to Invariant Manifolds

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Acta Mathematica Sinica,English Series 1996年第4期12卷 372-373,375-378页

作者： Zhu Deming Department of Mathematics East China Normal University Shanghai, 200062 China Department of Mathematics East China Normal University Shanghai China

Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic i... 详细信息

关键词： Invariant manifold Exponential dichotomy Heteroclinic orbit transversality

Absolute continuity of the distribution of some Markov geometric series

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Science China Mathematics 2007年第11期50卷 1521-1528页

作者： Ai-hua FAN~(1,2+) Ji-hong ZHANG~1 1 Department of Mathematics,Wuhan University,Wuhan 430072,China 2 LAMFA,UMR 6140 CNRS,Universitéde Picardie Jules Verne,33,Rue Saint Leu,80039 Amiens Cedex 1,France Department of Mathematics Wuhan University Wuhan China LAMFA UMR 6140 CNRS Université de Picardie Jules Verne Amiens Cedex 1 France

Let (■_n)≥0 be the Markov chain of two states with respect to the probability measure of the maximal entropy on the subshift spaceΣ_A defined by Fibonacci incident matrix *** consider the measureμ_λof the probability distribution of the random seriesΣ_(n=0)~∞■_nλ~n(0<λ<1).It is proved thatμ_λis singular ifλ∈(0,(5^(1/2)-1)/2) and thatμ_λis absolutely continuous for almost allλ∈((5^(1/2)-1)/2,0.739).

关键词： Markov series absolute continuity measure of maximal entropy dimension of a measure transversality

Non-periodic perturbations and transversal heteroclinic orbits

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Chinese Science Bulletin 1995年第9期40卷 715-718页

作者：朱德明 Department of Mathematics East China Normal University Shanghai 200062 China

We consider an n-dimensional system x=f(x)+εg(t,x,μ), (1)where f,g∈C_b^2 for x∈ΩR^n,t∈R,0<ε<<1,μ∈R^m,and Ω is some open set.

关键词： heteroclinic orbits transversality exponential dichotomy integral manifold.

MELNIKOV VECTOR WITH HIGHER ORDER

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Annals of Differential Equations 1996年第3期12卷 371-379页

作者： Zhu Deming(East China Normal University,200062)

By using the geometrical method, the higher order Melnikov vector and the associatedcriteria for the persistence, transversality and tangency of homoclinic and heteroclinic orbits are established. Examples of applicat... 详细信息

关键词： invariant manifold,tangent space heteroclinic orbit transversality exponential dichotomy.